Gockenbach M.S. Partial Differential Equations. Analytical and Numerical Methods
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608
Index
change
of
variables
in
a
multiple integral,
471
characteristic polynomial
of
a
second-order ODE,
82
of
a
square matrix,
69
characteristic root
of
a
second-order ODE,
82
Cholesky
factorization, 474,
496
chromium,
278
co-domain,
32
coefficients
discontinuous,
264
compatibility condition, 240, 243, 250,
276,
350
for
a
singular linear system,
45
complete
normed vector space,
452
complete orthogonal sequence,
449
completeness
of
^,490
of
L
2
,
453
complex numbers,
394
concentration
time evolution
of, 221
concentration gradient,
17
conditioning
of
a
matrix, 481,
486
conjugate directions,
485
conjugate gradients
algorithm,
485
convergence,
486
conservation
of
energy,
173
continuously
differentiable
function,
33
convergence
mean-square, 149,
419
pointwise,
419
uniform,
420
convolution,
424
periodic,
425
copper, 228, 243, 244, 265,
278
d'Alembert's
solution
to the
wave equa-
tion,
288
data
structure
for
describing
a
triangulation,
467
DCT,
414
delta
function,
126, 205,
318
sifting
property,
126
delta
sequence,
126
dense matrix,
186
density,
10, 330
descent direction,
479
determinant,
69
DFT,
404
differential
equation,
1
autonomous,
109
constant
coefficient,
4, 82, 91
converting
to a first-order
system,
79
first-order
linear,
87
homogeneous,
3, 202
inhomogeneous,
3, 96
linear,
2, 91
linear system,
260
nonconstant
coefficient,
4
nonlinear,
3
order,
2
ordinary,
1, 79, 82,
213,
260
partial,
1
right-hand side,
4
scalar,
5
system,
5, 91, 108
differential
operator
symmetric,
137
differentiating
an
integral,
11
diffusion,
16
diffusion
coefficient,
18
diffusion
equation,
18
inhomogeneous,
220
dimension,
50
Dirac
delta
function,
126, 205,
318
Dirichlet condition,
13
Dirichlet kernel,
423
discrete cosine transform,
414
discrete Fourier transform,
404
discrete sine transform,
410
and
the
FFT,
411
disk
Fourier series
on a,
359,
372
displacement
function,
21
divergence
operator,
329
608
change
of
variables
in
a multiple integral, 471
characteristic polynomial
of
a second-order ODE, 82
of
a square matrix, 69
characteristic
root
of
a second-order ODE, 82
Cholesky factorization, 474, 496
chromium, 278
co-domain, 32
coefficients
discontinuous, 264
compatibility condition, 240, 243, 250,
276,350
for a singular linear system, 45
complete
normed vector space, 452
complete orthogonal sequence, 449
completeness
of
HI,
490
of L2, 453
complex numbers, 394
concentration
time
evolution of,
221
concentration gradient, 17
conditioning
of a matrix, 481, 486
conjugate directions, 485
conjugate gradients
algorithm, 485
convergence, 486
conservation of energy, 173
continuously differentiable function, 33
convergence
mean-square, 149, 419
pointwise, 419
uniform, 420
convolution, 424
periodic, 425
copper, 228, 243, 244, 265, 278
d'Alembert's
solution
to
the
wave equa-
tion, 288
data
structure
for describing a triangulation, 467
DCT,
414
delta
function, 126, 205, 318
sifting property, 126
delta
sequence, 126
dense
matrix,
186
density, 10, 330
descent direction, 479
determinant, 69
DFT,
404
differential equation,
1
autonomous, 109
Index
constant
coefficient,
4,
82,
91
converting
to
a first-order system,
79
first-order linear, 87
homogeneous, 3, 202
inhomogeneous,
3,
96
linear,
2,
91
linear system, 260
nonconstant coefficient,
4
nonlinear, 3
order, 2
ordinary, 1, 79, 82, 213, 260
partial,
1
right-hand side, 4
scalar, 5
system,
5,
91, 108
differential
operator
symmetric, 137
differentiating
an
integral,
11
diffusion, 16
diffusion coefficient, 18
diffusion equation, 18
inhomogeneous, 220
dimension,
50
Dirac
delta
function, 126, 205, 318
Dirichlet condition, 13
Dirichlet kernel, 423
discrete cosine transform, 414
discrete Fourier transform, 404
discrete sine transform, 410
and
the
FFT,
411
disk
Fourier series
on
a, 359, 372
displacement function,
21
divergence
operator,
329
Index
609
theorem,
330
domain
of
a
function,
31
domain
of
dependence,
289
domain
of
influence,
290
dot
product,
45, 55, 71
DST,
410
eigenfunction,
68,
145,
494
complex-valued,
394
of
a
differential
operator,
139
eigenpairs
on
a
disk,
369
under Dirichlet conditions,
145
under
mixed boundary conditions,
151,
153
under Neumann conditions,
233
under periodic boundary condi-
tions,
247
eigenspace,
70
eigenvalue,
68,
116,
494
of
a
differential
operator,
139
of
a
symmetric
matrix,
139
eigenvalue problem
general,
455
generalized,
495
eigenvector,
68
of
a
symmetric matrix,
139
elastic
bar,
21
elliptic
regularity,
492
energy inner product, 182,
379
energy
norm,
182
Euclidean
n-space
complex,
395
Euler's
formula,
83, 394
Euler's method, 117, 261,
310
backward, 118, 263,
310
improved,
104
even
extension,
418
even
function,
415
and the
full
Fourier series,
417
existence,
238
of
solutions
to a
linear system,
38
explicit method,
118
extension
even,
418
odd,
417
periodic,
425
fast
Fourier transform, 393,
407
FFT, 393,
407
Fick's law,
17
fill-in, 477
finite
element method,
77,
167,
173
for
eigenvalue problems,
495
for
the
heat
equation,
256
for
the
wave equation,
306
Fourier
coefficients,
149
complex,
397
rate
of
decay,
436
sine,
149
Fourier series,
77,
149,
155
method
of, 166
complex, 397,
401
double
sine series,
343
for
the
wave equation,
291
full,
249, 252,
398
in
three dimensions,
355
on
a
rectangular domain,
339
pointwise
convergence,
429
sine,
149,
211
Fourier's law,
331
Fourier's
law of
heat
conduction,
12
Fredholm
alternative,
45, 241
free
nodes,
379
function,
31, 35
function
space,
33
fundamental
frequency,
497
of
a
circular drum,
375
of
a
vibrating string,
294
fundamental
theorem
of
calculus,
11,
23,
101, 327, 330,
338
Galerkin method, 173, 180, 193, 258,
269,
275, 307,
378
Gaussian elimination,
43, 77, 196
for
a
banded matrix,
476
operation count,
53,
474,
486
pseudo-code,
474
Index
theorem, 330
domain
of a function,
31
domain of dependence,
289
domain of influence, 290
dot product, 45, 55,
71
DST,410
eigenfunction, 68, 145, 494
complex-valued, 394
of a differential operator, 139
eigenpairs
on a disk, 369
under Dirichlet conditions, 145
under mixed boundary conditions,
151, 153
under Neumann conditions,
233
under periodic boundary condi-
tions, 247
eigenspace,
70
eigenvalue, 68, 116, 494
of a differential operator, 139
of a symmetric matrix, 139
eigenvalue problem
general,
455
generalized,
495
eigenvector,
68
of a symmetric matrix, 139
elastic
bar,
21
elliptic regularity, 492
energy inner product, 182,379
energy norm,
182
Euclidean n-space
complex, 395
Euler's formula, 83, 394
Euler's method, 117, 261, 310
backward, 118, 263, 310
improved, 104
even extension, 418
even function, 415
and the full Fourier series, 417
existence, 238
of solutions
to
a linear system, 38
explicit method, 118
extension
even, 418
odd, 417
periodic,
425
fast Fourier transform, 393, 407
FFT,
393, 407
Fick's law, 17
fill-in, 477
609
finite element method, 77, 167, 173
for eigenvalue problems, 495
for the heat equation, 256
for the wave equation, 306
Fourier coefficients, 149
complex, 397
rate
of decay, 436
sine, 149
Fourier series,
77,
149,
155
method
of,
166
complex, 397,
401
double sine series, 343
for the wave equation,
291
full, 249, 252, 398
in three dimensions, 355
on a rectangular domain, 339
pointwise convergence, 429
sine, 149,
211
Fourier's law, 331
Fourier's law of heat conduction,
12
Fredholm alternative, 45,
241
free nodes, 379
function, 31,
35
function space,
33
fundamental frequency, 497
of a circular drum, 375
of a vibrating string, 294
fundamental theorem of calculus, 11,
23,
101, 327, 330, 338
Galerkin method, 173, 180, 193, 258,
269,275,307,378
Gaussian elimination, 43,
77,
196
for a banded matrix, 476
operation count,
53,474,486
pseudo-code, 474
610
Index
general solution
of
the
wave equation,
286
generalized eigenvalue problem,
495
generalized
function,
126,
205
Gibbs's phenomenon, 148, 344, 435,
443
gold,
250
Gram-Schmidt procedure,
74,183,
457
gravitational constant,
27
gravity,
27
Green's
first
identity, 337, 339,
377
Green's
function,
77, 123
causal,
124
for
a
BVP,
203
for
a
PDE,
127
for
the
heat
equation,
279
grid,
101,
188
irregular,
111
guitar,
296,
318
H
l
,m
#<J,490
heat
capacity,
10, 19
heat
equation,
9, 12,
245,
275
and finite
elements,
257
in
a
bar,
333
in
three dimensions,
332
in
two
dimensions,
334
inhomogeneous,
13
weak
form,
257
heat
flow, 9
steady-state,
14
heat
flux, 11, 12, 330
homogeneous
versus heterogeneous,
13
Hooke's law,
22
Hookean,
22
IBVP,
14
implicit method,
118
initial condition,
14, 80,
246,
258
for
the
wave equation,
24
initial value problem
for
the
wave equation,
285
initial-boundary value problem,
14, 274
for
the
wave equation,
291
inner product,
56, 59, 181
complex,
395
complex
L
2
,
396
energy,
182
inner
product space,
56
integral
area,
329
iterated,
329
line,
329
surface,
329
volume,
328
integrating
an
ODE,
102
integrating factor,
87
integration
by
parts, 138, 178, 307,
336,
489
interpolant
piecewise linear,
492
inverse
of
a
differential
operator,
207
inverse discrete Fourier transform,
404
iron,
19,
214, 217, 222, 228, 230, 236,
244,
260, 264, 281, 323,
459
iterated
integral,
329
iterative
method
for
solving
a
linear system,
478
Jacobian
matrix,
471
jump discontinuity,
427
Kelvin,
19
kinetic energy,
174
L
2
,
445
L
2
inner product,
59, 60
Laplace's equation, 336,
352
Laplacian,
332
in
polar coordinates,
362
lead,
255
Lebesgue
integral,
446
measure,
446
line integral,
329
line search,
480
linear combination,
50
linear interpolant,
309
610
general solution
of
the
wave equation,
286
generalized eigenvalue problem, 495
generalized function, 126,
205
Gibbs's phenomenon, 148, 344, 435,
443
gold,250
Gram-Schmidt procedure,
74,
183,457
gravitational constant,
27
gravity,
27
Green's first identity, 337, 339, 377
Green's function,
77,
123
causal,124
for a
BVP,
203
for a PDE, 127
for the heat equation, 279
grid, 101, 188
irregular,
111
guitar, 296, 318
HI,
490
HJ,490
heat capacity, 10,
19
heat equation, 9, 12, 245,
275
and finite elements, 257
in a bar, 333
in three dimensions, 332
in two dimensions, 334
inhomogeneous,
13
weak form,
257
heat
flow,
9
steady-state,
14
heat flux, 11, 12, 330
homogeneous
versus heterogeneous,
13
Hooke's law,
22
Hookean,22
IBVP,
14
implicit method, 118
initial condition, 14, 80, 246, 258
for the wave equation,
24
initial value problem
for the wave equation,
285
initial-boundary value problem, 14,274
for the wave equation,
291
inner product, 56,
59,
181
complex,
395
complex L2,
396
energy,
182
inner product space,
56
integral
area, 329
iterated, 329
line, 329
surface, 329
volume, 328
integrating
an
ODE,
102
integrating factor, 87
Index
integration by parts, 138, 178, 307,
336,489
interpolant
piecewise linear,
492
inverse
of a differential operator, 207
inverse discrete Fourier transform, 404
iron, 19, 214, 217, 222, 228, 230, 236,
244, 260, 264, 281, 323, 459
iterated integral, 329
iterative method
for solving a linear system, 478
Jacobian matrix,
471
jump discontinuity, 427
Kelvin,
19
kinetic energy, 174
L
2
,445
L2
inner product,
59,
60
Laplace's equation, 336, 352
Laplacian, 332
in polar coordinates, 362
lead,255
Lebesgue
integral, 446
measure, 446
line integral, 329
line search, 480
linear combination,
50
linear interpolant, 309
Index
611
linear
operator,
31, 132
definition,
35
linear operator equation,
31, 36
linear system
algebraic,
31, 131
linearly independent,
52
load vector,
183
computation
of, 467
for
a
two-dimensional problem,
378
Lotka-Volterra
predator-prey model,
108
LU
factorization,
474
Maple,
147
mapping,
31
mass
matrix,
258, 265, 275, 308,
495
Mathematica,
147
MATLAB,
147
matrix,
31
banded, 386,
475
dense,
186
Gram,
62
identity,
46
inverse,
46
invertible,
46
mass, 308,
495
nonsingular,
46, 53
singular,
270
sparse, 173, 186, 189, 386,
473
stiffness,
308,
495
symmetric,
71, 72,
139,
272
tridiagonal, 188, 192,
272
mean value theorem,
431
mean-square,
60
mean-square convergence, 149,
419
of
Fourier series,
448
measurable
function,
446
set,
446
mesh,
188
method
of
lines
for
the
heat
equation,
260
middle
C,
295,
305
modulus
of
elasticity,
22
Young's,
22
multiplicity
of
eigenvalues,
69
natural
frequency
of
a
string, 293,
318
Neumann
condition,
13
Newton,
26
Newton's
law of
cooling
and
boundary conditions,
20
and the
heat
equation,
20
Newton's
second
law of
motion,
23, 29
nodal values,
190
nodes,
188
nondimensionalization,
217
norm,
59
energy,
182
normal derivative,
332
normal equations,
62
normal modes
of
a
vibrating string,
293
null
space,
134
of
a
matrix,
42
trivial,
42
odd
extension,
417
odd
function,
415
and the
full
Fourier series,
417
ODE,
1
operator,
31, 35
derivative,
36
differential,
36, 132
linear,
132
matrix,
35, 131
nonlinear,
35, 49
wave,
285
order
of
a
differential
equation,
2
of
a
numerical
method,
103
orthogonal basis,
183
orthogonality
of
functions,
60
of
vectors,
56
partial
derivative
weak,
490
partial
pivoting,
473
Index
linear operator, 31, 132
definition,
35
linear operator equation, 31, 36
linear system
algebraic, 31,
131
linearly independent,
52
load vector, 183
computation
of,
467
for a two-dimensional problem, 378
Lotka-Volterra predator-prey model,
108
L U factorization, 474
Maple, 147
mapping,
31
mass matrix, 258, 265, 275, 308, 495
Mathematica, 147
MATLAB, 147
matrix,
31
banded,
386,475
dense, 186
Gram,
62
identity, 46
inverse,
46
invertible, 46
mass, 308, 495
nonsingular, 46,
53
singular, 270
sparse, 173, 186, 189, 386, 473
stiffness, 308, 495
symmetric, 71, 72, 139,
272
tridiagonal, 188, 192,
272
mean value theorem,
431
mean-square, 60
mean-square convergence, 149, 419
of Fourier series, 448
measurable
function, 446
set, 446
mesh, 188
method of lines
for
the
heat equation, 260
middle C, 295, 305
modulus
of elasticity,
22
Young's,
22
multiplicity
of eigenvalues,
69
natural
frequency
of a string, 293, 318
Neumann condition, 13
Newton,
26
Newton's law of cooling
611
and
boundary conditions,
20
and
the
heat
equation, 20
Newton's second law of motion, 23,
29
nodal values, 190
nodes, 188
nondimensionalization, 217
norm,
59
energy, 182
normal derivative, 332
normal equations,
62
normal modes
of a vibrating string,
293
null space, 134
of a matrix, 42
trivial, 42
odd extension, 417
odd function, 415
and
the
full Fourier series, 417
ODE, 1
operator, 31, 35
derivative,
36
differential, 36, 132
linear, 132
matrix, 35,
131
nonlinear, 35, 49
wave, 285
order
of a differential equation, 2
of a numerical method, 103
orthogonal basis, 183
orthogonality
of functions, 60
of vectors,
56
partial derivative
weak, 490
partial pivoting, 473
612
Index
Pascal,
26
PDE,
1
periodic convolution,
425
periodic
extension,
425
periodic function,
415
piano,
298
piece
wise
continuous,
427
piecewise
linear
approximation,
492
basis
function,
190, 257,
269
function,
188,
189
interpolant,
259
piecewise polynomials,
173
piecewise
smooth,
427
plate
iron,
459
pointwise
convergence,
419
of
a
complex Fourier series,
429
of
a
Fourier cosine series,
433
of
a
Fourier sine series,
433
of
a
full
Fourier series,
432
Poisson's equation,
335
on
a
disk,
373
polar coordinates,
360
potential energy,
173
elastic, 173,
174
external,
175
gravitational,
175
minimal,
177
power
series solution
of
Bessel's equation,
364
preconditioned conjugate gradients,
486
pressure,
25
prime factorization
and
the
FFT,
407
principle
of
virtual work,
177
product rule,
337
programming
finite
elements,
386
projection theorem, 182,
397
Pythagorean theorem,
55, 428
quadrature,
102
choosing
a
rule,
470
general
formula,
469
one-point Gauss rule,
469
one-point rule
on a
triangle,
470
over
an
arbitrary triangle,
471
three-point rule
for a
triangle,
470
quarter-wave
cosine series,
153
quarter-wave
sine
functions,
162
quarter-wave
sine series,
151
range
of
a
function,
31
of
a
linear operator,
38
resonance, 295, 305, 318,
319
Riemann
integral,
446
Riesz
representation theorem,
490
RK4,
107
Runge-Kutta method,
106
Runge-Kutta-Fehlberg
method,
111
scalar,
32, 395
semidiscretization
in
space,
260
separation
of
variables, 225, 340,
348
in
polar coordinates,
362
shifting
the
data,
164, 197, 222, 299,
321
sifting
property,
205
silver,
255
Simpson's rule,
106
sink
heat,
12
snapshot
temperature,
215
Sobolev
space,
490
source
heat,
12
span,
52
sparse matrix, 186, 386,
473
sparsity
pattern, 386,
477
specific
heat,
10, 330
and
temperature,
19
spectral method,
156
for
a
BVP,
144
for
a
system
of
ODEs,
96
for
solving
a
linear system,
167
speed
of
a
wave,
288
square wave,
433
stability,
118
of
a
time-stepping scheme,
310
612
Pascal,
26
PDE,
1
periodic convolution, 425
periodic extension, 425
periodic function,
415
piano,
298
piecewise continuous, 427
piecewise linear
approximation, 492
basis function, 190, 257,
269
function, 188, 189
interpolant,
259
piecewise polynomials, 173
piecewise smooth, 427
plate
iron, 459
pointwise convergence, 419
of a complex Fourier series, 429
of a Fourier cosine series, 433
of a Fourier sine series, 433
of a full Fourier series, 432
Poisson's equation, 335
on a disk, 373
polar coordinates, 360
potential energy, 173
elastic, 173,
174
external, 175
gravitational,
175
minimal, 177
power series solution
of Bessel's equation, 364
preconditioned conjugate gradients, 486
pressure,
25
prime factorization
and
the
FFT,
407
principle of virtual work, 177
product rule, 337
programming finite elements, 386
projection theorem, 182, 397
Pythagorean theorem, 55, 428
quadrature,
102
choosing a rule, 470
general formula, 469
one-point Gauss rule, 469
one-point rule on a triangle, 470
Index
over an arbitrary triangle,
471
three-point rule for a triangle, 470
quarter-wave cosine series,
153
quarter-wave sine functions, 162
quarter-wave sine series,
151
range
of a function,
31
of a linear operator, 38
resonance, 295, 305, 318, 319
Riemann integral, 446
Riesz representation theorem, 490
RK4,
107
Runge-Kutta method, 106
Runge-Kutta-Fehlberg method,
111
scalar, 32, 395
semidiscretization in space,
260
separation of variables, 225, 340, 348
in polar coordinates, 362
shifting the data, 164, 197, 222, 299,
321
sifting property,
205
silver,
255
Simpson's rule,
106
sink
heat,
12
snapshot
temperature,
215
Sobolev space, 490
source
heat,
12
span,
52
sparse matrix, 186, 386, 473
sparsity pattern, 386, 477
specific heat, 10, 330
and temperature,
19
spectral method,
156
for a BVP, 144
for a system of ODEs,
96
for solving a linear system, 167
speed
of a wave,
288
square wave, 433
stability, 118
of a time-stepping scheme, 310
Index
613
stability region,
116
stainless
steel,
26
standing wave,
294
steady-state
and
Neumann conditions,
237
temperature,
225
steel,
302
steepest descent
algorithm,
481
direction,
481
stiff
differential
equation, 117,
260
stiff
system
of
ODEs,
117
stiffness
of
a
bar,
22
stiffness
matrix, 183, 191, 258, 265,
275,
308,
495
algorithm
for
computing,
466
computation
of, 465
for
a
two-dimensional problem,
378
null
space,
271
singular, 270,
275
strain,
22
string
elastic,
27,
131,
160
sagging,
30
strong
form
of
a
BVP, 177,
267
subspace,
34
of
R
n
,
39
superposition,
287
principle
of, 47, 85
support
of
a
function,
192,
382
surface
integral,
329
symmetry
and
eigenvalues,
72
of
a
differential
operator,
137
of
a
matrix,
71
of
the
Laplacian,
337
temperature gradient,
12
test
function, 178, 267,
306
thermal
conductivity,
12, 19, 330
time
step,
101
time-stepping,
101
trace
theorem,
490
transient behavior,
117,
118
transpose
and the dot
product,
71
of
a
matrix,
45
trapezoidal rule,
402
triangulation, 379,
382
data
structure for,
467
description
of, 462
tridiagonal matrix, 188,
192
trigonometric interpolation,
406
uniform
convergence,
420
of
a
complex Fourier series,
439
of
a
Fourier cosine
series,
441
of
a
Fourier sine series,
441
of
a
full
Fourier
series,
441
of
continuous
functions,
443
of
cosine series,
441
uniqueness,
238
of
solutions
to a
linear system,
42
unit impulse,
126
unit normal
vector,
328
unit point source,
126
variation
of
parameters,
98
variational
form
of
a
BVP,
173
vector,
31, 32
function
as, 132
vector
field, 329
vector space,
132
complex,
395
definition,
32
vibrating string,
292
virtual
work,
177
volume
integral,
328
wave
left-moving,
288
right-moving,
288
wave
equation,
23, 173
acoustic,
334
d'Alembert's solution,
288
general solution,
286
homogeneous,
293
in an
infinite
domain,
285
Index
stability region, 116
stainless steel,
26
standing wave, 294
steady-state
and Neumann conditions, 237
temperature,
225
steel, 302
steepest descent
algorithm,
481
direction,
481
stiff differential equation, 117,
260
stiff system of ODEs, 117
stiffness
of a bar,
22
stiffness matrix, 183, 191, 258, 265,
275, 308, 495
algorithm for computing, 466
computation
of,
465
for a two-dimensional problem, 378
null space,
271
singular, 270,
275
strain,
22
string
elastic,
27,
131, 160
sagging, 30
strong form
of a BVP, 177, 267
subspace, 34
of
Rn,
39
superposition, 287
principle
of,
47,
85
support
of a function, 192, 382
surface integral, 329
symmetry
and eigenvalues,
72
of a differential operator, 137
of a matrix,
71
of the Laplacian, 337
temperature gradient,
12
test function, 178, 267, 306
thermal conductivity, 12, 19, 330
time step,
101
time-stepping,
101
trace theorem, 490
transient behavior, 117, 118
transpose
and
the dot product,
71
of a matrix,
45
trapezoidal rule,
402
triangulation, 379, 382
data
structure for, 467
description of, 462
tridiagonal matrix, 188, 192
trigonometric interpolation, 406
uniform convergence, 420
613
of a complex Fourier series, 439
of a Fourier cosine series,
441
of a Fourier sine series,
441
of a full Fourier series,
441
of continuous functions, 443
of cosine series,
441
uniqueness, 238
of solutions to a linear system,
42
unit impulse, 126
unit normal vector, 328
unit point source, 126
variation of parameters, 98
variational form
of a
BVP, 173
vector, 31, 32
function as, 132
vector field, 329
vector space, 132
complex, 395
definition, 32
vibrating string, 292
virtual work, 177
volume integral, 328
wave
left-moving, 288
right-moving, 288
wave equation, 23, 173
acoustic, 334
d'Alembert's solution, 288
general solution, 286
homogeneous, 293
in an infinite domain,
285
in
three dimensions,
334
in
two
dimensions,
334
inhomogeneous,
296
on
a
disk,
374
on
a
square,
346
with
a
point source,
318
wave
operator,
285
wave
speed, 288,
296
weak
derivative,
490
weak
form
ofaBVP,
173,
177,266,
267
of
a BVP in two or
three dimen-
sions,
378
of
the
wave equation,
307
Young's modulus,
22
614
Index
614
in three dimensions, 334
in two dimensions, 334
inhomogeneous,
296
on a disk, 374
on a square, 346
with a point source, 318
wave operator,
285
wave speed, 288,
296
weak derivative, 490
weak form
of a BVP, 173, 177, 266, 267
of a
BVP in two or three dimen-
sions, 378
of
the
wave equation, 307
Young's modulus,
22
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